742 problems

Let $L(s)$ range over the several families of $L$-functions considered by Keating and Snaith, evaluated at the central point $s=1/2$. Let $\mathrm{SO}(2N)$ and $\mathrm{USp}(2N)$ d…

Density conjecture. The Katz–Sarnak philosophy predicts

In quantum chaos, consider a quantum system whose classical dynamics are sufficiently chaotic. Bohigas–Giannoni–Schmit conjecture. Its local spectral statistics should be described…

Generic quantum integrable systems are quantum systems with integrable classical dynamics, and locally Poissonian level statistics means that their unfolded neighbouring energy lev…

Hilbert–Pólya-type conjecture. The nontrivial zeros of $\zeta(s)$ should be realized as eigenvalues of a self-adjoint operator; the coherence Hamiltonian provides an analogue whose…

A random matrix has local spectral properties determined by its symmetry class, while its entry distribution does not affect them. Wigner–Dyson–Mehta universality conjecture. The l…

Airy edge conjecture. The edge limit of the additions should still be the Airy $beta$-process under suitable rescaling, even when some $kappa_l$ are negative.

Let $\lambda(\alpha,\beta)$ denote the eigenvalue process defined by the paper's Dyson SDE, with interaction parameter $\alpha$ and confinement parameter $\beta$. For the time-resc…

Let $\rho=\beta+i\gamma$ range over the non-trivial zeros of the Riemann zeta function, and define … Let $G$ denote Barnes' $G$-function, defined for $s\in\mathbb{C}$ by … For…

Baik–Deift–Johansson conjecture. As $n\to\infty$, the joint distribution of the first few rows of $\lambda$, after the proper shift and rescaling, is the same as the distribution o…

Let a quantum system be chaotic, and consider the local features of its spectrum. Random matrix theory (RMT) is the statistical framework relating such spectral features to random…

The quartic Kontsevich model is a model whose correlation functions are studied through topological recursion and intersection numbers on moduli spaces of curves. Blobbed topologic…

Let a symmetric palindromic Toeplitz matrix have a limiting eigenvalue-gap distribution, and let a real symmetric random-matrix ensemble have independent matrix elements drawn from…

Wigner-Dyson-Gaudin-Mehta conjecture. The local spectral statistics of Wigner matrices exhibit universality: they are independent of the underlying distribution of the matrix entri…

Let $q=p^a$ be a prime power, let $\mathbb{F}_q[T]$ be the polynomial ring over the finite field $\mathbb{F}_q$, and let $\mathfrak{M}_2(2g)$ denote the set of primitive quadratic…

A GOE matrix is a real symmetric random matrix with Gaussian entries up to the orthogonal symmetry constraint; more generally, Hermitian random matrices belong to the unitary symme…

Let $A$ be the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices, let $H=A/\sqrt{d-1}$, and consider the bulk eigenvalues of $H$. Fixed-degree random-…

Let the tridiagonal random matrix models of Dumitriu and Edelman be scaled appropriately so that they approximate random stochastic operators. Edelman–Sutton conjecture. Scaling li…

Let $E(s)$ denote the probability that the interval $(x^*-s,x^*+s)$ is empty, and suppose that the density of states behaves as $|x-x^*|^{\alpha}$ as $x\to x^*$. Forrester–Chen–Eri…

Let a family of $L$-functions be given, and consider the logarithms of their central values. Keating–Snaith conjecture. The logarithm of the central values of $L$-functions in vari…

Keating–Rodgers–Roditty-Gershon–Rudnick's variance conjecture. When $Q^{1+\varepsilon}<X<Q^{k-\varepsilon}$ and $X\to\infty$,

Let a quantum system arise as the quantization of a classically chaotic system, and consider its eigenvalues and their mean spacing. Random Matrix conjecture. Generically, the eige…

Tight-bound conjecture. The tight upper bound for this quantity is actually $p^5/n^4$.

The Riemann zeta function is denoted by $\zeta$. The zeros of $\zeta$ are the complex numbers at which it vanishes, and a large Hermitian random matrix is a random matrix whose eig…

Let $\Sigma_1$ and $\Sigma_2$ be the covariance matrices, and let $H$ be their joint limiting spectral distribution. For any $n_1,n_2\in\mathbb{N}$ and every selection of non-negat…